A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

Journal Article · 26 October 2017 · SIAM Journal on Scientific Computing

DOI: https://doi.org/10.1137/16M1075582 · OSTI ID:1598333

Lee, Kookjin ^[1]; Elman, Howard C. ^[1]

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1. Univ. of Maryland, College Park, MD (United States)

Herein, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a coarse-grid method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the present method is displayed by numerical experiments on benchmark problems.

• Research Organization: Univ. of Maryland, College Park, MD (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
• Grant/Contract Number: SC0009301; DMS-1418754
• OSTI ID: 1598333
• Journal Information: SIAM Journal on Scientific Computing, Vol. 39, Issue 5; ISSN 1064-8275
• Publisher: SIAMCopyright Statement
• Country of Publication: United States
• Language: English